| L(s) = 1 | + (−1.67 − 1.18i)2-s + (−1.91 + 1.74i)3-s + (0.738 + 2.09i)4-s + (0.374 − 0.564i)5-s + (5.27 − 0.653i)6-s + (3.47 − 3.58i)7-s + (0.124 − 0.437i)8-s + (0.341 − 3.68i)9-s + (−1.29 + 0.503i)10-s + (3.13 − 1.44i)11-s + (−5.06 − 2.72i)12-s + (0.578 + 2.03i)13-s + (−10.0 + 1.88i)14-s + (0.269 + 1.73i)15-s + (2.73 − 2.19i)16-s + (−3.82 − 0.473i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.839i)2-s + (−1.10 + 1.00i)3-s + (0.369 + 1.04i)4-s + (0.167 − 0.252i)5-s + (2.15 − 0.266i)6-s + (1.31 − 1.35i)7-s + (0.0440 − 0.154i)8-s + (0.113 − 1.22i)9-s + (−0.410 + 0.159i)10-s + (0.945 − 0.435i)11-s + (−1.46 − 0.785i)12-s + (0.160 + 0.563i)13-s + (−2.69 + 0.504i)14-s + (0.0694 + 0.447i)15-s + (0.683 − 0.549i)16-s + (−0.927 − 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.437075 - 0.208532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.437075 - 0.208532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (10.1 + 0.0108i)T \) |
| good | 2 | \( 1 + (1.67 + 1.18i)T + (0.664 + 1.88i)T^{2} \) |
| 3 | \( 1 + (1.91 - 1.74i)T + (0.276 - 2.98i)T^{2} \) |
| 5 | \( 1 + (-0.374 + 0.564i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (-3.47 + 3.58i)T + (-0.215 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-3.13 + 1.44i)T + (7.15 - 8.35i)T^{2} \) |
| 13 | \( 1 + (-0.578 - 2.03i)T + (-11.0 + 6.84i)T^{2} \) |
| 17 | \( 1 + (3.82 + 0.473i)T + (16.4 + 4.14i)T^{2} \) |
| 19 | \( 1 + (-6.28 + 2.00i)T + (15.5 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.195 + 2.11i)T + (-22.6 + 4.22i)T^{2} \) |
| 29 | \( 1 + (-3.48 - 0.214i)T + (28.7 + 3.56i)T^{2} \) |
| 31 | \( 1 + (6.28 + 2.43i)T + (22.9 + 20.8i)T^{2} \) |
| 37 | \( 1 + (1.64 - 1.02i)T + (16.4 - 33.1i)T^{2} \) |
| 41 | \( 1 + (-1.23 - 1.86i)T + (-15.9 + 37.7i)T^{2} \) |
| 43 | \( 1 + (0.755 - 0.405i)T + (23.7 - 35.8i)T^{2} \) |
| 47 | \( 1 + (0.872 + 1.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.24 + 5.67i)T + (-48.1 - 22.1i)T^{2} \) |
| 59 | \( 1 + (-10.3 - 10.6i)T + (-1.81 + 58.9i)T^{2} \) |
| 61 | \( 1 + (5.89 - 7.80i)T + (-16.6 - 58.6i)T^{2} \) |
| 67 | \( 1 + (-2.91 + 0.732i)T + (59.0 - 31.6i)T^{2} \) |
| 71 | \( 1 + (4.14 - 0.255i)T + (70.4 - 8.72i)T^{2} \) |
| 73 | \( 1 + (4.88 - 9.81i)T + (-43.9 - 58.2i)T^{2} \) |
| 79 | \( 1 + (0.255 + 0.513i)T + (-47.6 + 63.0i)T^{2} \) |
| 83 | \( 1 + (-7.72 - 1.94i)T + (73.1 + 39.2i)T^{2} \) |
| 89 | \( 1 + (14.8 - 2.77i)T + (82.9 - 32.1i)T^{2} \) |
| 97 | \( 1 + (0.980 - 0.121i)T + (94.0 - 23.6i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.66149994523763372594399683323, −11.67038663716942298579839938321, −11.33182008952403017369616536823, −10.62246084653565452108180343461, −9.627155214780607979051708446830, −8.661893806303069512518296374743, −7.13412740809728633564051018965, −5.23693973398408024407752748519, −4.08097751936900105150219199440, −1.15654539802406408850379304514,
1.57136585079981336320984516862, 5.32746369248605023280440216331, 6.26055642607615234492775302030, 7.28435244514216470696732834557, 8.298670059128364581525696540154, 9.333938243138560827804511784710, 10.86330033468444967295601010387, 11.82427143009704807949121521557, 12.53895890321845249800104528931, 14.22377147455939408402205576320
